If X and Y are two sets such that X has 40 elements , `XuuY` has 50 elements and X `cap` Y has 20 elements, how many elements does Y have?

Answer

To find the number of elements in set Y, we can use the formula for the union of two sets:

\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

Where:
- \( n(X \cup Y) \) is the number of elements in the union of sets X and Y.
- \( n(X) \) is the number of elements in set X.
- \( n(Y) \) is the number of elements in set Y.
- \( n(X \cap Y) \) is the number of elements in the intersection of sets X and Y.

Given:
- \( n(X) = 40 \)
- \( n(X \cup Y) = 50 \)
- \( n(X \cap Y) = 20 \)

Now, we can substitute the known values into the formula:

1. Start with the formula:
\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

2. Substitute the known values:
\[ 50 = 40 + n(Y) - 20 \]

3. Simplify the equation:
\[ 50 = 40 - 20 + n(Y) \]
\[ 50 = 20 + n(Y) \]

4. Isolate \( n(Y) \):
\[ n(Y) = 50 - 20 \]
\[ n(Y) = 30 \]

Thus, the number of elements in set Y is **30**.